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Theorem ablgrp 12889
Description: An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
Assertion
Ref Expression
ablgrp  |-  ( G  e.  Abel  ->  G  e. 
Grp )

Proof of Theorem ablgrp
StepHypRef Expression
1 isabl 12888 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simplbi 274 1  |-  ( G  e.  Abel  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2146   Grpcgrp 12738  CMndccmn 12884   Abelcabl 12885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-abl 12887
This theorem is referenced by:  ablgrpd  12890  ablinvadd  12909  ablsub2inv  12910  ablsubadd  12911  ablsub4  12912  abladdsub4  12913  abladdsub  12914  ablpncan2  12915  ablpncan3  12916  ablsubsub  12917  ablsubsub4  12918  ablpnpcan  12919  ablnncan  12920  ablnnncan  12922  ablnnncan1  12923  ablsubsub23  12924
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